Quaternion distinguished generic representations of $\mathrm{GL}_{2n}$

Abstract: Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}m(D)$ can be regarded as a subgroup of $\mathrm{GL}{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}m(D)$-distinguished representations of $\mathrm{GL}{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group $\mathrm{U}_{2m}$ to be $\mathrm{GL}_m(D)$-distinguished.